Question

Dada a função f(x) = ax + b e sabendo-se que f(3) = 5 e f(-2) = -5 calcule f(½)

Answer 1

Answer:

$f(1/2)=0$

Step-by-step explanation:

Given this function:

$f(x)=ax+b$

We need to find the value of:

$f(1/2)$

This is really simple because the problem give us the following data:

$f(3)=5\\\\and\\\\f(-2)=-5$

Using this data we can write a 2x2 system of equations like this:

Let:

$f(3)=5=a(3)+b\\\\\rightarrow3a+b=5\hspace{10}(1)$

and

$f(-2)=5=a(-2)+b\\\\\rightarrow-2a+b=-5\hspace{10}(2)$

Our system will be:

$3a+b=5\hspace{28}(1)\\-2a+b=-5\hspace{10}(2)$

We can solve this system using different methods, however I will use elimination method because I believe it is really simple. So:

$(1)-(2)\\\\5-(-5)=3a-(-2a)+b-b\\\\10=5a\\\\Solving\hspace{3}for\hspace{3}a\\\\a=\frac{10}{5} =2$

Replacing $a$ into (1)

$3(2)+b=5\\\\6+b=5\\\\Solving\hspace{3}for\hspace{3}b\\\\b=5-6=-1$

Now we found the values of $a$ and $b$, our function is given by:

$f(x)=2x-1$

Let's verify it:

$f(3)=2(3)-1=6-1=5$

$f(-2)=2(-2)-1=-4-1=-5$

As you can see the function satisfies the data provided by the problem. So we can conclude that it is correct.

Finally, let's find $f(x)$ for $x=1/2$ :

$f(1/2)=2(\frac{1}{2} )-1=\frac{2}{2} -1=1-1=0$

Hence:

$f(1/2)=0$